physics’1b electricity’&’magne4sm’ · 2012-03-07 · inductors’ " answer!"...
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Physics 1B Electricity & Magne4sm
Frank Wuerthwein (Prof) Edward Ronan (TA)
UCSD
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Outline of today
• Inductors
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Quiz 4 Content
• Chapter 23 • This includes everything we have done in class including all of today’s slides.
• It also implies that you need to be able to analyze circuits with R and L as we do today.
• So strictly speaking, you s4ll need to know some of the things you learned in earlier chapters in order to analyze these circuits.
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Inductors • In a circuit, capacitors store electric fields for later use.
• Also, for a circuit we have inductors which produce useable magne4c fields.
• The symbol for an inductor is:
" If a solenoid of N windings has a current, I, moving through it that creates a magnetic flux, ΦB, through its center, then we define it to have an inductance, L, of:
!
L =N"B
I
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Inductors • Inductance is basically the total magne4c flux per unit of current required to produce it.
• Like capacitance is the charge per unit voltage. • The inductance of an inductor depends only on its geometry (just like capacitors).
• The unit of inductance is the Henry (H):
!
1 H =1 T "m2
A" So, for a solenoid with length, l, N windings, loop
area, A, and current I:
!
L =N BA( )I
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Inductors But, from before, we kmow that for a solenoid:
So the inductance of a solenoid is independent of the applied current.
To increase inductance for a coil:
increase the number of windings, N, OR increase the area per loop, A, OR decrease the length of the solenoid, l.
!
B = µoN!( )I
" This means that inductance for a solenoid becomes:
!
L =N µoNI
!" # $ %
& ' A
I=
µoN2A!
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Self Inductance • If the current in a solenoid is changing in 4me, then its magne4c flux will change, leading to an induced emf.
• This is called self-‐induc4on.
" But recall from Faraday’s Law:
!
L =N"B
I
!
LI = N"B
!
" = #N $%B
$t
!
" = #$ LI( )$t
" Since L is only dependent on geometry:
!
"L = #L $I$t
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Inductance • Recall that resistance was a measure of how much a certain element opposed current flow.
• Inductance can be interpreted as a measure of opposi4on to the rate of change in the current.
" As a circuit is completed (switch closed), the current begins to increase, but the inductor produces an emf that opposes the increasing current.
" There is a voltage drop across the inductor.
IL
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Inductance • Thus, the current doesn’t change from 0 to its maximum value instantaneously.
• If you are increasing the current, I, in an inductor, there will be increasing magne4c flux inside the inductor.
" The inductor will then create an emf to oppose this change in current.
" When the current reaches its maximum value, the rate of change is zero, and, thus, emf by inductor is also zero.
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Inductance • When, a circuit is connected, it takes a certain amount of 4me for the current to reach its final value.
" The time constant, τ, for an RL circuit is the time required for the current in the circuit to reach 63.2% of its final value.
!
"L =LR
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Inductance The current at any 4me can be found by:
" If you are decreasing the current, I, in an inductor, there will be decreasing magnetic flux inside the inductor.
" The inductor will then create an emf to oppose this change in current.
!
I ="R1# e
# t$ L
% & '
( ) *
IL
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Inductors • Example • A solenoid having an inductance of 6.30μH is connected in series with a 1.20kΩ resistor. If a 14.0V babery is connected across the pair, how long will it take for the current through the resistor to reach 80.0% of its final value?
" Answer " This is a basic RL circuit connected to an emf source. " We should turn to the circuit equation from last slide.
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Inductors " Answer
" where ɛ/R is the maximum value that the current will attain. Since we want 80% of this value we can replace it with:
!
I ="R1# e
# t$ L
% & '
( ) *
" Now, we just have to solve for time:
!
0.80( )Imax = Imax 1" e"t# L
$ % &
' ( )
!
0.80( ) = 1" e" t# L
$ % &
' ( )
!
e"t# L =1" 0.80 = 0.20
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Inductors " Answer
" Note that the current gets to nearly its maximum value rather quickly.
" Also note that even though the potential difference of the battery was given, it didn’t affect the overall answer.
!
ln e" t# L
$ % &
' ( ) = "t # L
= ln 0.20( )
!
t = "LR#
$ %
&
' ( ln 0.20( ) = "
6.30 )10"6H1.20 )103*#
$ %
&
' ( "1.609( )
!
t = 8.45 "10#9s
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Inductors • Example • The figure below shows a circuit that contains three iden4cal resistors with resistance R = 9.0Ω, two iden4cal inductors with inductance L = 2.0mH, and an ideal babery with emf ɛ = 18V. What is the current through the babery long aier the switch has been closed?
" Answer " First, you must define a
direction for the current I through the battery.
" Up makes sense.
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Inductors " Answer
" There will be voltage drops at the inductors and the resistors given by:
" In circuits with a battery and RL circuits, the current will eventually reach some final value.
" This means that it will not be changing in time and, thus, there will be no voltage drop across the inductors.
" This means that the inductors will act as regular wires when the current reaches a steady value.
" Yet there will still be voltage drop across the resistors.
!
"VR = IR
!
"VL = #L "I"t
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Inductors " Answer
" This means that the circuit can be reduced to the following a long time after the switch is closed:
" We basically have three resistors in parallel which we can easily solve by equivalent resistance.
!
1Req
=1R
+1R
+1R
=3R
!
Req =R3
=9"3
= 3"
" The current will be:
!
I ="VR
Req
=18V3#
= 6.0A
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Energy in an Inductor • We say that a capacitor can store energy in its electric field which is given by:
!
ECap = 12C "V( )2
" We also say that an inductor can store energy in its magnetic field which is given by:
!
EL = 12 LI
2
" If you attempt to change the current through the inductor, the energy stored in the inductor will perform work to oppose this change.
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Concept Ques4on The figures to the right show three circuits with iden4cal baberies, inductors, and resistors. Rank the circuits according to the current through the babery a long 4me aier the switch is closed (greatest first).
A) 1, 2, 3.
B) 1, 3, 2.
C) 2, 1, 3.
D) 2, 3, 1.
E) 3, 2, 1.
2 1
3
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For Next Time (FNT) " Finish reading Chapter 23
" Finish working on the homework for Chapter 23